Symmetric group on finite set has Sylow subgroup of prime order
- Bertrand's postulate: The version we use states that for any , there exists a prime such that .
- Sylow subgroups exist
The symmetric group of degree has order . By fact (1), there exists a prime such that . Thus, the largest power of dividing is . In particular, this means that the order of the -Sylow subgroup (which exists by fact (2)) is .